Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Opinion Dynamics with Multiple Adversaries

Akhil Jalan, Marios Papachristou

TL;DR

This work extends opinion-dynamics modeling by allowing multiple adversaries to strategically misreport intrinsic opinions, inducing potentially greater polarization and disagreement than single-adversary settings. Using the Friedkin–Johnsen framework, the authors derive a Pure Strategy Nash Equilibrium for manipulated intrinsic opinions and quantify social inefficiency via the Price of Misreporting, providing worst-case bounds tied to the Laplacian spectrum. They validate the theory on Twitter, Reddit, and Polblogs datasets, showing that strategic manipulation can markedly alter network-wide outcomes and proposing efficient detection and identifications approaches based on hypothesis testing and robust regression. The proposed learning algorithms enable platforms to detect manipulation and recover the set of deviators, offering practical tools for governance that emphasize structural monitoring and transparency to mitigate strategic distortions. Overall, the paper delivers a rigorous game-theoretic treatment of multi-adversary manipulation in networked opinion dynamics and actionable methods for platform defense.

Abstract

Opinion dynamics models how the publicly expressed opinions of users in a social network coevolve according to their neighbors as well as their own intrinsic opinion. Motivated by the real-world manipulation of social networks during the 2016 US elections and the 2019 Hong Kong protests, a growing body of work models the effects of a strategic actor who interferes with the network to induce disagreement or polarization. We lift the assumption of a single strategic actor by introducing a model in which any subset of network users can manipulate network outcomes. They do so by acting according to a fictitious intrinsic opinion. Strategic actors can have conflicting goals, and push competing narratives. We characterize the Nash Equilibrium of the resulting meta-game played by the strategic actors. Experiments on real-world social network datasets from Twitter, Reddit, and Political Blogs show that strategic agents can significantly increase polarization and disagreement, as well as increase the "cost" of the equilibrium. To this end, we give worst-case upper bounds on the Price of Misreporting (analogous to the Price of Anarchy). Finally, we give efficient learning algorithms for the platform to (i) detect whether strategic manipulation has occurred, and (ii) learn who the strategic actors are. Our algorithms are accurate on the same real-world datasets, suggesting how platforms can take steps to mitigate the effects of strategic behavior.

Opinion Dynamics with Multiple Adversaries

TL;DR

This work extends opinion-dynamics modeling by allowing multiple adversaries to strategically misreport intrinsic opinions, inducing potentially greater polarization and disagreement than single-adversary settings. Using the Friedkin–Johnsen framework, the authors derive a Pure Strategy Nash Equilibrium for manipulated intrinsic opinions and quantify social inefficiency via the Price of Misreporting, providing worst-case bounds tied to the Laplacian spectrum. They validate the theory on Twitter, Reddit, and Polblogs datasets, showing that strategic manipulation can markedly alter network-wide outcomes and proposing efficient detection and identifications approaches based on hypothesis testing and robust regression. The proposed learning algorithms enable platforms to detect manipulation and recover the set of deviators, offering practical tools for governance that emphasize structural monitoring and transparency to mitigate strategic distortions. Overall, the paper delivers a rigorous game-theoretic treatment of multi-adversary manipulation in networked opinion dynamics and actionable methods for platform defense.

Abstract

Opinion dynamics models how the publicly expressed opinions of users in a social network coevolve according to their neighbors as well as their own intrinsic opinion. Motivated by the real-world manipulation of social networks during the 2016 US elections and the 2019 Hong Kong protests, a growing body of work models the effects of a strategic actor who interferes with the network to induce disagreement or polarization. We lift the assumption of a single strategic actor by introducing a model in which any subset of network users can manipulate network outcomes. They do so by acting according to a fictitious intrinsic opinion. Strategic actors can have conflicting goals, and push competing narratives. We characterize the Nash Equilibrium of the resulting meta-game played by the strategic actors. Experiments on real-world social network datasets from Twitter, Reddit, and Political Blogs show that strategic agents can significantly increase polarization and disagreement, as well as increase the "cost" of the equilibrium. To this end, we give worst-case upper bounds on the Price of Misreporting (analogous to the Price of Anarchy). Finally, we give efficient learning algorithms for the platform to (i) detect whether strategic manipulation has occurred, and (ii) learn who the strategic actors are. Our algorithms are accurate on the same real-world datasets, suggesting how platforms can take steps to mitigate the effects of strategic behavior.

Paper Structure

This paper contains 35 sections, 9 theorems, 28 equations, 8 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Characterizing Nash Equilibria with Multiple Adversaries.
  3. Real-World Experiments to Understand Properites of Equilibria.
  4. Analysis of Equilibrium Outcomes Under Different Sets of Strategic Actors.
  5. Learning Algorithms for the Platform.
  6. Preliminaries and Notations
  7. Related Work
  8. Opinion Dynamics
  9. Disagreement and Polarization in Social Networks
  10. Manipulation of Dynamic Games.
  11. Learning from Strategic Data.
  12. Real-world Datasets
  13. (1) Twitter dataset ($n = 548, m = 3638$).
  14. (2) Reddit dataset ($n = 556, m = 8969$).
  15. (3) Political Blogs (Polblogs) dataset ($n = 1490, m = 16178$).
  16. ...and 20 more sections

Key Result

Theorem 2.2

Let $\mathbb T_i = (1 - \alpha_i) (B^T L_i B) + \alpha_i (B^T \bm{e}_i \bm{e}_i^T B) \in \mathbb{R}^{n \times n}$ and $\bm{y}_i = \alpha_i B_{ii} \bm{s}_i$. The Nash equilibria, if any exist, are given by solutions $s^\prime \in \mathbb{R}^n$ to the following constrained linear system:

Figures (8)

  • Figure 1: Visualization of the strategic equilibrium ($\bm{z}^\prime$) on the Karate Club Graph for two different choices of $S$. The truthful intrinsic opinions have been taken to be $\bm{s} = \bm{u}_2$ where $\bm{u}_2$ is the Fiedler eigenvector of $G$. The white nodes correspond to the nodes in $S$. For the other nodes, the nodes colored in blue (resp. red) correspond to nodes whose public opinion $\bm{z}^\prime_i$ increased (resp. decreased), i.e., $(\bm{z}^\prime_i - \bm{z}_i) / \bm{z}_i \ge 0$ (resp. $(\bm{z}^\prime_i - \bm{z}_i) / \bm{z}_i < 0$) after $\bm{s}'$ was chosen.
  • Figure 2: Plot of truthful intrinsic opinions ($s$) and strategic opinions ($s'$), and truthful public opinions ($z$) compared to the strategic public opinions ($z'$) for the nodes belonging to $S$. $S$ is taken to be the top-50% in terms of their eigenvector centrality. In both cases we have taken $\alpha_i \in \{ 0.25, 0.5 \}$ for all nodes. We fit a linear regression between $s'$ and $s$ (resp. between $z$ and $z'$). We report the effect size $\theta$ which corresponds to the slope of the linear regression and the $P$-value with respect to the null hypothesis ($\theta = 0$). $^{***}$ stands for $P < 0.001$, $^{**}$ stands for $P < 0.01$ and $^{*}$ stands for $P < 0.05$.
  • Figure 3: Strategic misreports on the Polblogs dataset, where $S$ consists of the top 50% of agents by eigenvector centrality. Nodes are labeled as liberal ($\bm{s}_i = -1$) or conservative ($\bm{s}_i = +1$). A node is said to change belief if $\bm{z}_i$ and $\bm{z}_i'$ have opposite signs. In scatterplots (a), (c), (d), and (e), point shape indicates whether a belief change occurs, while color denotes the intrinsic opinion. We observe more frequent belief changes among liberal blogs than conservative ones (panel (b)). Panels (c) and (d) report truthful and strategic public opinions as functions of $\log \bm{\pi}_i$, while panel (e) shows the absolute change $|\bm{z}_i' - \bm{z}_i|$. Regression analysis reveals significant effects of $\log \bm{\pi}_i$ ($^{***}\!:\,P<0.001$, $^{**}\!:\,P<0.01$, $^{*}\!:\,P<0.05$; coefficients $\theta$) on $\bm{z}$, $\bm{z}'$, and $|\bm{z}' - \bm{z}|$, consistent with power-law structure. Finally, relative belief changes are more dispersed among liberal sources than conservative ones (panel (f)).
  • Figure 4: Polarization ratio ($\mathcal{P}(z')/\mathcal{P}(z)$), disagreement ratio ($\mathcal{D}(z') / \mathcal{D}(z)$), and price of misreporting ($C(z') / C(z)$) for the three datasets for varying susceptibility to persuasion values. We have set all susceptibilities $\alpha_i$ to the same value $\alpha$. The Twitter dataset has the largest variation in all three ratios compared to the others. $S$ is taken to be the top-50% nodes in terms of their eigenvector centrality.
  • Figure 5: Polarization ratio ($\mathcal{P}(\bm{z}^\prime)/\mathcal{P}(\bm{z})$), disagreement ratio ($\mathcal{D}(\bm{z}^\prime) / \mathcal{D}(\bm{z})$), and price of misreporting ($C(\bm{z}^\prime) / C(\bm{z})$) for the three datasets for varying the size of $|S|$. The size of $|S|$ corresponds to the top $p$ percent of the actors ($|S| = \lceil p n \rceil$) based on their eigenvector centrality (in decreasing order), for $p \in [0.01, 0.1]$. The susceptibility parameter is set to $\alpha_i = 0.5$.
  • ...and 3 more figures

Theorems & Definitions (11)

  • Definition 2.1: Instrinsic belief lying problem.
  • Theorem 2.2: Nash Equilibrium
  • Corollary 2.3
  • Corollary 2.4: Pure Strategy Nash Equilibria
  • Theorem 3.1
  • Definition 4.1: SSC and SSS Conditions
  • Proposition 4.2
  • Theorem 4.3: torrent-2015
  • Proposition 4.4
  • Proposition 4.5
  • ...and 1 more